Defining and Analyzing New Classes Associated with (λ,γ)-Symmetrical Functions and Quantum Calculus
Abstract
:1. Introduction
- , where a and b are real (or complex) constants.
- , introduced and studied by Darus et al. [16].
- = , the class motivated by Agrawal and Sahoo in [17].
- = , the well-known class of starlike function of order by Robertson [18].
- , motivated by Sakaguchi [19].
- = , which was first introduced by Ismail et al. [9].
- , which reduces to the well-known class defined by Janowski [1].
- , the class introduced by Nevanlinna [20].
- In the basic case with the definition simplifies, since there are no roots of unity involved. Let , . In this case, the condition becomes the following expression:Consider the following function:We compute its q-derivative as follows:Simplifying the above expression can verify the condition. For this specific example, if the subordination holds, belongs to .
- In the case of symmetric points with , we use the primitive 2nd roots of unity .Let , , and . In this case, the condition becomes the following expression:Consider the following function:We construct as follows:Then, the following is true:Simplifying this and checking the subordination condition verifies that satisfies the condition for .
- In the general case with , we use the primitive 3rd roots of unity Let , , and . In this case, the condition becomes the following expression:Consider the following function:We construct as follows:Computing and and averaging them can verify the function’s symmetry properties.
2. Main Results
3. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Louati, H.; Al-Rezami, A.Y.; Darem, A.A.; Alsarari, F. Defining and Analyzing New Classes Associated with (λ,γ)-Symmetrical Functions and Quantum Calculus. Mathematics 2024, 12, 2603. https://doi.org/10.3390/math12162603
Louati H, Al-Rezami AY, Darem AA, Alsarari F. Defining and Analyzing New Classes Associated with (λ,γ)-Symmetrical Functions and Quantum Calculus. Mathematics. 2024; 12(16):2603. https://doi.org/10.3390/math12162603
Chicago/Turabian StyleLouati, Hanen, Afrah Y. Al-Rezami, Abdulbasit A. Darem, and Fuad Alsarari. 2024. "Defining and Analyzing New Classes Associated with (λ,γ)-Symmetrical Functions and Quantum Calculus" Mathematics 12, no. 16: 2603. https://doi.org/10.3390/math12162603